Singular boundary value problems for nonlinear elliptic equations in nonsmooth domains

Abstract

Let $\Omega$ be a piecewise-regular domain in $\Bbb R^N$ and O an irregular point on its boundary $\partial \Omega$. We study under what conditions on $q$ any solution $u$ of (E) $-\Delta u + g(x,u)=0$ where $g$ has a $q$-power-like growth at infinity ($q>1$) which coincides on $\partial \Omega \setminus {\{\text{O}}\}$ with a continuous function defined on whole $\partial \Omega$, can be extended as a continuous function in $\bar \Omega$.